Chemically synthesized quantum dots (QDs) can potentially enable new classes of highly flexible, spectrally tunable lasers processible from solutions. Despite a considerable progress over the past years, colloidal-QD lasing, however, is still at the laboratory stage and an important challenge - realization of lasing with electrical injection - is still unresolved. A major complication, which hinders the progress in this field, is fast nonradiative Auger recombination of gain-active multicarrier species such as trions (charged excitons) and biexcitons. Recently, we explored several approaches for mitigating the problem of Auger decay by taking advantage of a new generation of core/multi-shell QDs with a radially graded composition that allow for considerable (nearly complete) suppression of Auger recombination by "softening" the electron and hole confinement potentials. Using these specially engineered QDs, we have been able to realize optical gain with direct-current electrical pumping, which has been a long-standing goal in the field of colloidal nanostructures. Further, we apply these dots to practically demonstrated the viability of a "zero-threshold-optical-gain" concept using not neutral but negatively charged particles wherein the pre-existing electrons block either partially or completely ground-state absorption. Such charged QDs are optical-gain-ready without excitation and, in principle, can exhibit lasing at vanishingly small pump levels. All of these exciting recent developments demonstrate a considerable promise of colloidal nanomaterials for implementing solution-processible optically and electrically pumped laser devices operating across a wide range of wavelengths and fabricated on virtually any substrate using a variety of optical-cavity designs.